Optical communication systems are known in which optical signals carrying data are transmitted from a first node to a second or receive node over an optical fiber. At the receive node, the optical signals are converted into corresponding electrical signals, which are then further processed.
The receive node typically includes a receiver. One type of receiver used is a coherent receiver. In the coherent receiver, the received signal is mixed with an output of a local oscillator (LO) in an optical hybrid circuit, the outputs of which are made incident on photodetectors to generate analog electrical output signals. The analog signals are sampled at a sample rate by analog-to-digital converter (ADC) circuits configurable to supply outputs for further processing by a Digital Signal Processor (DSP), for example. Thus, for the coherent receiver, the amplitude, phase, and state of polarization of the optical signal are all transferred to the electrical domain for digital signal processing in the DSP.
The optical signals may be subject to various impairments including chromatic dispersion (CD) and polarization mode dispersion (PMD), etc. CD is due to various frequency components in each signal traveling at different velocities. PMD is due to various polarization components in each signal traveling at different velocities. As a result of the impairments, the in-phase (I) and quadrature (Q) components of electronic signals representative of data carried by the optical signals may have different delay, frequency response, and polarization characteristics. Such differences may be resolved and both CD and PMD compensation can be achieved electronically using an equalizer in the receiver.
The equalizer may be a finite-impulse response (FIR) digital filter, for example. Such filters include inputs or taps, and symbols are transferred from one tap to the next, multiplied by a coefficient at each tap, and the resulting products are summed. Typically, the coefficients are selected based on parameters, such as estimates of CD, PMD, etc., and such estimates may be obtained by “training” the equalizer. According to one known method of training, the transmit node sends a known training sequence of bits or periodically inserts a known sequence of bits in the transmitted signal at certain intervals and transmits the signal to the receiver. The receiver recognizes the known training sequence and uses it to train the equalizer using known methods.
A disadvantage of sending a known sequence of bits during start up of an optical communication system is the requirement for an external control between the transmitter and receiver nodes, which increases complexity of the system. A further disadvantage of periodically inserting a known sequence of bits in the transmission signal is that the insertion increases the overall bit rate. An increase in the overall bit rate typically requires higher bandwidth components which increases cost.
An alternative method for training which does not include sending or inserting a known sequence of bits is known as “blind equalization”. According to the blind equalization method, the training is performed as a function of the measuring of the actual received signal output from the equalizer. One known blind equalization method is a constant modulus algorithm (CMA). (For ready reference, CMA as used herein to refer to “the constant modulus algorithm”). This algorithm is also referred to as the Godard algorithm since its origin is from a reference authored by Godard (IEEE Transactions on Communications, Vol. COM-28, No. 11, 1980, pp. 1867-1875). In accordance with CMA, filter coefficients are selected through an iterative process (“training”), whereby a cost function is associated with the modulus or magnitude of signals output from the equalizer are equalized to a fixed value (e.g., 1). CMA is effective in training filters to provide equalization of signals that have a certain type of symmetry, i.e., E{an2}=0, where E is the expectation and an are the data symbols. Such signals include quadrature phase shift keying (QPSK) modulated signals, in which signal phase is modulated to have one of four values separated from one another by 90°. For polarization multiplexed signals, methods such as the one reported by Vgenis (IEEE Photonics Technology Letters, Vol. 22, No. 1., 2010, pp. 45-47) can be added to train each of the equalizers for each polarization.
In some known systems, the equalizer receives signals indicative of a binary phase shift keying (BPSK) modulation format in which data is conveyed by modulating the signal phase to have one of two phases separated by 180°. For the BPSK format, an in the above equation takes on values of +1 or −1 with a probability of 1/2. Consequently, the formula E{an2} for BPSK evaluates to 1, not 0. Thus, in practice, if CMA is used in connection with a BPSK signal, the trained output of the equalizer may or may not be BPSK. What is needed, therefore, is a method, system, and apparatus that will enable CMA to be used reliably for blind equalization, including for systems where BPSK modulation is employed.